Theorem lspsolv 21426

Description: If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lspsolv.v	⊢ 𝑉 = (Base‘𝑊)
lspsolv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspsolv.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspsolv	⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Proof of Theorem lspsolv

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsolv.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		lspsolv.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspsolv.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		eqid 2760	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2760	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2760	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
7		eqid 2760	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
8		eqid 2760	. . 3 ⊢ {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)} = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)}
9		lveclmod 21386	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	9	adantr 486	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑊 ∈ LMod)
11		simpr1 1213	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝐴 ⊆ 𝑉)
12		simpr2 1214	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ 𝑉)
13		simpr3 1215	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
14	13	eldifad 4051	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14	lspsolvlem 21425	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
16	4	lvecdrng 21385	. . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
17	16	ad2antrr 739	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (Scalar‘𝑊) ∈ DivRing)
18		simprl 783	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
19	10	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑊 ∈ LMod)
20	12	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ 𝑉)
21		eqid 2760	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
22		eqid 2760	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
23	1, 4, 7, 21, 22	lmod0vs 21175	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
24	19, 20, 23	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
25	24	oveq2d 7436	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)(0g‘𝑊)))
26	11	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ 𝑉)
27	20	snssd 4743	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑌} ⊆ 𝑉)
28	26, 27	unssd 4190	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑌}) ⊆ 𝑉)
29	1, 3	lspssv 21263	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑌}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
30	19, 28, 29	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
31	30	ssdifssd 4083	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)) ⊆ 𝑉)
32	13	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
33	31, 32	sseldd 3909	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ 𝑉)
34	1, 6, 22	lmod0vrid 21173	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
35	19, 33, 34	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
36	25, 35	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = 𝑋)
37	36, 32	eqeltrd 2860	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
38	37	eldifbd 4052	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
39		simprr 785	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
40		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
41	40	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)))
42	41	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → ((𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) ↔ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
43	39, 42	syl5ibcom 248	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
44	43	necon3bd 2969	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) → 𝑟 ≠ (0g‘(Scalar‘𝑊))))
45	38, 44	mpd 16	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ≠ (0g‘(Scalar‘𝑊)))
46		eqid 2760	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
47		eqid 2760	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
48		eqid 2760	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
49	5, 21, 46, 47, 48	drnginvrl 21019	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
50	17, 18, 45, 49	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
51	50	oveq1d 7435	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
52	5, 21, 48	drnginvrcl 21016	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
53	17, 18, 45, 52	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
54	1, 4, 7, 5, 46	lmodvsass 21167	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
55	19, 53, 18, 20, 54	syl13anc 1399	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
56	1, 4, 7, 47	lmodvs1 21170	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
57	19, 20, 56	syl2anc 596	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
58	51, 55, 57	3eqtr3d 2803	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
59	33	snssd 4743	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑋} ⊆ 𝑉)
60	26, 59	unssd 4190	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ 𝑉)
61	1, 2, 3	lspcl 21256	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
62	19, 60, 61	syl2anc 596	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
63	1, 4, 7, 5	lmodvscl 21158	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
64	19, 18, 20, 63	syl3anc 1398	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
65		eqid 2760	. . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊)
66	1, 6, 65	lmodvpncan 21195	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
67	19, 64, 33, 66	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
68	1, 6	lmodcom 21188	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
69	19, 64, 33, 68	syl3anc 1398	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
70		ssun1 4176	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑋})
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ (𝐴 ∪ {𝑋}))
72	1, 3	lspss 21264	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑋})) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
73	19, 60, 71, 72	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
74	73, 39	sseldd 3909	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
75	69, 74	eqeltrd 2860	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
76	1, 3	lspssid 21265	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
77	19, 60, 76	syl2anc 596	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
78		snidg 4617	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
79		elun2 4181	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐴 ∪ {𝑋}))
80	33, 78, 79	3syl 19	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝐴 ∪ {𝑋}))
81	77, 80	sseldd 3909	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
82	65, 2	lssvsubcl 21224	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})) ∧ 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
83	19, 62, 75, 81, 82	syl22anc 852	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
84	67, 83	eqeltrrd 2861	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
85	4, 7, 5, 2	lssvscl 21235	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
86	19, 62, 53, 84, 85	syl22anc 852	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
87	58, 86	eqeltrrd 2861	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
88	15, 87	rexlimddv 3169	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149  {csn 4584  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433  ._rcmulr 17434  Scalarcsca 17436   ·_𝑠 cvsca 17437  0_gc0g 17615  -_gcsg 19151  1_rcur 20412  inv_rcinvr 20622  DivRingcdr 20985  LModclmod 21140  LSubSpclss 21211  LSpanclspn 21251  LVecclvec 21382

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8246  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-0g 17617  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-grp 19152  df-minusg 19153  df-sbg 19154  df-cmn 20001  df-abl 20002  df-mgp 20366  df-rng 20380  df-ur 20413  df-ring 20466  df-oppr 20572  df-dvdsr 20592  df-unit 20593  df-invr 20623  df-drng 20987  df-lmod 21142  df-lss 21212  df-lsp 21252  df-lvec 21383

This theorem is used by:  lssacsex  21427  lspsnat  21428  lsppratlem1  21430  lsppratlem3  21432  lsppratlem4  21433  lbsextlem4  21444  lindsenlbs  22162  lindsadd  38629